Challenging integration problem of an exponential function

While preparing a tutorial session for my students, I come across a very challenging integration, which is :$$\int_{-\infty}^\infty e^{t^2 }\ \mathrm{d}t$$

I attempted many methods to solve it but with no use. Your assistance will be greatly appreciated.

• Can you do $\int t e^{t^2} dt$. – Z Ahmed Jan 16 at 16:01
• Depends on how you interpret $\int$. If it means $\int_{-\infty}^\infty$, then this is a well-known result. If you looking for an antiderivative, then your tutorial session is not good... – user9464 Jan 16 at 16:05
• Thank you for our response . well it is integration from - infinity to + infinity ! what is the well-know result you referred to ? by the way it is tutorial on linear system analysis – Safa Jan 16 at 16:11
• Then you should add those bounds in your integral. See en.wikipedia.org/wiki/… – user9464 Jan 16 at 16:35
• Great ! This is what I was looking for , thanks alot – Safa Jan 16 at 17:15

Assuming you are looking for the antiderivative of $$e^{t^{2}}$$: $$\int e^{t^2 }\ \mathrm{d}t = \frac{\sqrt{\pi}}{2}\operatorname{erfi}(x),$$where $$\operatorname{erfi}(x)$$ is the imaginary error function defined as $$\operatorname{erfi}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{t^{2}}\mathrm{d}t.$$ There is no way of expressing the antiderivative of $$e^{t^{2}}$$ in terms of elementary functions.